Shall I get the HPI check myself or is it something the dealer did it and will have details of?
Dealers normally do them as a matter of course, and by all means ask to see it. If you ask them if the car has ever been damaged or written off and they confirm that it hasn't, get them to put it in writing at the point of purchase and you should be fine. If you don't ask the question, it can be a very grey area.
I wouldn't worry about doing a check yourself if buying from an established dealer.
Buying private is another matter. Unless the car is only a few hundred pounds, it's worth the few quid it costs doing a check... if only to make sure there is no finance or "logbook loans" outstanding. Logbook loans are not legally enforceable if a car is sold, but that doesn't stop them sending heavies round to put the wind up the new keeper.
If the dealer has the full logbook, it's worth having a look at it, but this applies more to private sales where the seller certainly should have the logbook in their possession.
If the car has ever been an insurance write off, it is clearly declared on the front page. If all is okay, all you should see is one line, "1. Declared new at first registration". Also check the issue date matches the date of last change of owner - if it doesn't, then the logbook has been re-issued to the current owner. They may have lost the original, or they may have handed it to a logbook loan company and applied for a replacement.
Also if I did find corrosion underneath is that a complete deal breaker or just a way to get the price down?
It depends how bad it is. "Seaside cars" can be terrible, and can give major headaches later on, but you never know for sure where they have lived. The place of registration is only that, where it was originally registered.
If corrosion is excessive, it is usually mentioned on an MOT, and brake pipes are normally the first safety-related items that show signs. It's not that uncommon on the rear pipes, but the MOT also mentions a front pipe too.
If you can have a look, or get someone to have a look under the rear, you should expect to see large suspension components with lots of surface rust, and there will be some rust on the body edges at that age, but most of the underbody paint and underseal coating should still be visible. If you see more rust than paint on the body, you have a problem.